Find All Complex Solutions 3x^(16/5)-192x^2=0

$3 x^{\frac{16}{5}} - 192 x^{2} = 0$

Find a common factor $x^{2}$ that is present in each term.

$3 {(x^{2})}^{\frac{8}{5}} - 192 x^{2}$

Substitute $u$ for $x^{2}$ .

$3 {(u)}^{\frac{8}{5}} - 192 u = 0$

Solve for $u$ .

Factor the left side of the equation.

Factor $3 u$ out of $3 u^{\frac{8}{5}} - 192 u$ .

Factor $3 u$ out of $3 u^{\frac{8}{5}}$ .

$3 u (u^{\frac{3}{5}}) - 192 u = 0$

Factor $3 u$ out of $- 192 u$ .

$3 u (u^{\frac{3}{5}}) + 3 u (- 64) = 0$

Factor $3 u$ out of $3 u (u^{\frac{3}{5}}) + 3 u (- 64)$ .

$3 u (u^{\frac{3}{5}} - 64) = 0$

Rewrite $u^{\frac{3}{5}}$ as ${(u^{\frac{1}{5}})}^{3}$ .

$3 u ({(u^{\frac{1}{5}})}^{3} - 64) = 0$

Rewrite $64$ as $4^{3}$ .

$3 u ({(u^{\frac{1}{5}})}^{3} - 4^{3}) = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = u^{\frac{1}{5}}$ and $b = 4$ .

$3 u ((u^{\frac{1}{5}} - 4) ({(u^{\frac{1}{5}})}^{2} + u^{\frac{1}{5}} \cdot 4 + 4^{2})) = 0$

Factor.

Simplify.

Multiply the exponents in ${(u^{\frac{1}{5}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3 u ((u^{\frac{1}{5}} - 4) (u^{\frac{1}{5} \cdot 2} + u^{\frac{1}{5}} \cdot 4 + 4^{2})) = 0$

Combine $\frac{1}{5}$ and $2$ .

$3 u ((u^{\frac{1}{5}} - 4) (u^{\frac{2}{5}} + u^{\frac{1}{5}} \cdot 4 + 4^{2})) = 0$

Move $4$ to the left of $u^{\frac{1}{5}}$ .

$3 u ((u^{\frac{1}{5}} - 4) (u^{\frac{2}{5}} + 4 u^{\frac{1}{5}} + 4^{2})) = 0$

Raise $4$ to the power of $2$ .

$3 u ((u^{\frac{1}{5}} - 4) (u^{\frac{2}{5}} + 4 u^{\frac{1}{5}} + 16)) = 0$

Reorder terms.

$3 u ((u^{\frac{1}{5}} - 4) (4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16)) = 0$

Remove unnecessary parentheses.

$3 u (u^{\frac{1}{5}} - 4) (4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$u^{\frac{1}{5}} - 4 = 0$

$4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16 = 0$

Set $u$ equal to $0$ .

$u = 0$

Set $u^{\frac{1}{5}} - 4$ equal to $0$ and solve for $u$ .

Set $u^{\frac{1}{5}} - 4$ equal to $0$ .

$u^{\frac{1}{5}} - 4 = 0$

Solve $u^{\frac{1}{5}} - 4 = 0$ for $u$ .

Add $4$ to both sides of the equation.

$u^{\frac{1}{5}} = 4$

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(u^{\frac{1}{5}})}^{5} = 4^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(u^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${(u^{\frac{1}{5}})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u^{\frac{1}{5} \cdot 5} = 4^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$u^{\frac{1}{5} \cdot 5} = 4^{5}$

Rewrite the expression.

$u^{1} = 4^{5}$

Simplify.

$u = 4^{5}$

Simplify the right side.

Raise $4$ to the power of $5$ .

$u = 1024$

Set $4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16$ equal to $0$ and solve for $u$ .

Set $4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16$ equal to $0$ .

$4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16 = 0$

Solve $4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16 = 0$ for $u$ .

Find a common factor $u^{\frac{1}{5}}$ that is present in each term.

$4 u^{\frac{1}{5}} {(u^{\frac{1}{5}})}^{2} + 16$

Substitute $u$ for $u^{\frac{1}{5}}$ .

$4 (u) {(u)}^{2} + 16 = 0$

Solve for $u$ .

Multiply $u$ by ${(u)}^{2}$ by adding the exponents.

Move ${(u)}^{2}$ .

$4 ({(u)}^{2} u) + 16 = 0$

Multiply ${(u)}^{2}$ by $u$ .

Raise $u$ to the power of $1$ .

$4 ({(u)}^{2} u^{1}) + 16 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 u^{2 + 1} + 16 = 0$

Add $2$ and $1$ .

$4 u^{3} + 16 = 0$

Subtract $16$ from both sides of the equation.

$4 u^{3} = - 16$

Add $16$ to both sides of the equation.

$4 u^{3} + 16 = 0$

Factor $4$ out of $4 u^{3} + 16$ .

Factor $4$ out of $4 u^{3}$ .

$4 (u^{3}) + 16 = 0$

Factor $4$ out of $16$ .

$4 u^{3} + 4 \cdot 4 = 0$

Factor $4$ out of $4 u^{3} + 4 \cdot 4$ .

$4 (u^{3} + 4) = 0$

Divide each term in $4 (u^{3} + 4) = 0$ by $4$ and simplify.

Divide each term in $4 (u^{3} + 4) = 0$ by $4$ .

$\frac{4 (u^{3} + 4)}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (u^{3} + 4)}{4} = \frac{0}{4}$

Divide $u^{3} + 4$ by $1$ .

$u^{3} + 4 = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$u^{3} + 4 = 0$

Subtract $4$ from both sides of the equation.

$u^{3} = - 4$

Take the cube root of both sides of the equation to eliminate the exponent on the left side.

$u = \sqrt[3]{- 4}$

Simplify $\sqrt[3]{- 4}$ .

Rewrite $- 4$ as ${(- 1)}^{3} \cdot 4$ .

Rewrite $- 4$ as $- 1 (4)$ .

$u = \sqrt[3]{- 1 (4)}$

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$u = \sqrt[3]{{(- 1)}^{3} \cdot 4}$

Pull terms out from under the radical.

$u = - 1 \sqrt[3]{4}$

Rewrite $- 1 \sqrt[3]{4}$ as $- \sqrt[3]{4}$ .

$u = - \sqrt[3]{4}$

Substitute $u$ for $u$ .

$u^{\frac{1}{5}} = - \sqrt[3]{4}$

Solve for $u$ .

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

${(u^{\frac{1}{5}})}^{5} = {(- \sqrt[3]{4})}^{5}$

Simplify the exponent.

Simplify the left side.

Simplify ${(u^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${(u^{\frac{1}{5}})}^{5}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$u^{\frac{1}{5} \cdot 5} = {(- \sqrt[3]{4})}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$u^{\frac{1}{5} \cdot 5} = {(- \sqrt[3]{4})}^{5}$

Rewrite the expression.

$u^{1} = {(- \sqrt[3]{4})}^{5}$

Simplify.

$u = {(- \sqrt[3]{4})}^{5}$

Simplify the right side.

Simplify ${(- \sqrt[3]{4})}^{5}$ .

Apply the product rule to $- \sqrt[3]{4}$ .

$u = {(- 1)}^{5} {\sqrt[3]{4}}^{5}$

Raise $- 1$ to the power of $5$ .

$u = - {\sqrt[3]{4}}^{5}$

Rewrite ${\sqrt[3]{4}}^{5}$ as ${(4^{5})}^{\frac{1}{3}}$ .

$u = - \sqrt[3]{4^{5}}$

Raise $4$ to the power of $5$ .

$u = - \sqrt[3]{1024}$

Rewrite $1024$ as $8^{3} \cdot 2$ .

Factor $512$ out of $1024$ .

$u = - \sqrt[3]{512 (2)}$

Rewrite $512$ as $8^{3}$ .

$u = - \sqrt[3]{8^{3} \cdot 2}$

Pull terms out from under the radical.

$u = - (8 \sqrt[3]{2})$

Multiply $8$ by $- 1$ .

$u = - 8 \sqrt[3]{2}$

The final solution is all the values that make $3 u (u^{\frac{1}{5}} - 4) (4 u^{\frac{1}{5}} + u^{\frac{2}{5}} + 16) = 0$ true.

$u = 0, 1024, - 8 \sqrt[3]{2}$

Substitute $x$ for $u$ .

$x^{2} = 0, 1024, - 8 \sqrt[3]{2}$

Solve for $x^{2} = 0$ for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Solve for $x^{2} = 1024$ for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1024}$

Simplify $\pm \sqrt{1024}$ .

Rewrite $1024$ as $32^{2}$ .

$x = \pm \sqrt{32^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 32$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 32$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 32$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 32, - 32$

Solve for $x^{2} = - 8 \sqrt[3]{2}$ for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 8 \sqrt[3]{2}}$

Simplify $\pm \sqrt{- 8 \sqrt[3]{2}}$ .

Rewrite $- 8 \sqrt[3]{2}$ as ${(2 i)}^{2} \cdot (2 \sqrt[3]{2})$ .

Factor $- 4$ out of $- 8$ .

$x = \pm \sqrt{- 4 (2) \sqrt[3]{2}}$

Rewrite $- 4$ as ${(2 i)}^{2}$ .

$x = \pm \sqrt{{(2 i)}^{2} \cdot 2 \sqrt[3]{2}}$

Add parentheses.

$x = \pm \sqrt{{(2 i)}^{2} \cdot (2 \sqrt[3]{2})}$

Pull terms out from under the radical.

$x = \pm 2 i \sqrt{2 \sqrt[3]{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 i \sqrt{2 \sqrt[3]{2}}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 i \sqrt{2 \sqrt[3]{2}}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 i \sqrt{2 \sqrt[3]{2}}, - 2 i \sqrt{2 \sqrt[3]{2}}$

List all of the solutions.

$x = 0, 32, - 32, 2 i \sqrt{2 \sqrt[3]{2}}, - 2 i \sqrt{2 \sqrt[3]{2}}$

Find All Complex Solutions 3x^(16/5)-192x^2=0

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